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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Vector Algebra
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Q)

If $\overrightarrow a\:\:and\:\:\overrightarrow b$ are unit non collinear vectors and if $\overrightarrow u=\overrightarrow a-(\overrightarrow a.\overrightarrow b)\overrightarrow b$ and $\overrightarrow v=\overrightarrow a\times\overrightarrow b$ then $ |\overrightarrow v|= ?$

$\begin{array}{1 1} |\overrightarrow u| \\|\overrightarrow u|+|\overrightarrow u.\overrightarrow a| \\ |\overrightarrow u|+|\overrightarrow u\times\overrightarrow b| \\ |\overrightarrow u|+\overrightarrow u.(\overrightarrow a+\overrightarrow b) \end{array} $

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A)
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  • $\overrightarrow a\times (\overrightarrow b\times\overrightarrow c)=(\overrightarrow a.\overrightarrow c) \overrightarrow b-(\overrightarrow a.\overrightarrow b)\overrightarrow c$
Given: $|\overrightarrow a|=|\overrightarrow b|=1$
Also given that $\overrightarrow u=\overrightarrow a-(\overrightarrow a.\overrightarrow b)\overrightarrow b$
$\Rightarrow\: \overrightarrow u=(\overrightarrow b.\overrightarrow b)\overrightarrow a-(\overrightarrow a.\overrightarrow b)\overrightarrow b$ (Since $|\overrightarrow b|=1)$
$\Rightarrow\:\overrightarrow u=\overrightarrow b\times(\overrightarrow a\times\overrightarrow b)$
$\Rightarrow\:\overrightarrow u=\overrightarrow b\times \overrightarrow v$ (Since it is given that $\overrightarrow v=\overrightarrow a\times\overrightarrow b$)
$\Rightarrow\:|\overrightarrow u|=|\overrightarrow b||\overrightarrow v|sin\large\frac{\pi}{2}$
(Since $\overrightarrow v=\overrightarrow a\times\overrightarrow b\:\:is\:\:\perp\:\:to\:\:\overrightarrow b$)
$\Rightarrow\:|\overrightarrow u|=|\overrightarrow v|$ (Since $ |\overrightarrow b|=1$)
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